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I read around about Hermitian operators recently. Even though I understood their "essential" or "mathematical" role, I didn't quite get what they physically mean, if they mean anything at all.

What I understood was (correct me wherever I go wrong): when we talk about the classical world, any measurements that we make are completely real and have no complex component to them. In other words, all observables are just complex conjugates of their own values meaning they are real. Quantum observables on the other hand are a bit different. They’re represented by linear operators, that too of a special kind - Hermitian operators. These operators represent quantum observables that are equal to their own Hermitian conjugate.

Could someone explain the actual meaning of these words? Any idea to spark some sort of train of thought in me to lead me to the actual meaning behind this rather bookish (yet, I assume, complete) definition?

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    $\begingroup$ Welcome here it is a good question for sure, also difficult i think. The first thing is: that notion about measurements are real numbers and thus eigenvalues need to be real and so on, I think it's just something instructors say without thinking about it to much. If I want to I can Measure an angle and write it down as a complex number and there is no "real" (haha) problem with that. Of course in QM it's part of the formalism to have real observables, so there it would be wrong. To the actual question I'll leave that in more capable hands. $\endgroup$
    – Kuhlambo
    Commented Aug 8, 2022 at 11:23
  • $\begingroup$ I think this may be enlightening physics.stackexchange.com/a/264439/71413 $\endgroup$
    – Kuhlambo
    Commented Aug 8, 2022 at 11:47

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If I understand your question correctly, you are asking what is the intuitive meaning of the complex numbers appearing in quantum mechanics, when the classical world appears to deal only in real numbers.

There is an alternative approach to quantum mechanics that has been slowly developing over the past few decades using Geometric Algebra that claims to provide an intuitive, geometric, real-number picture of quantum mechanics that has proved very enlightening when it comes to developing a physical intuition for these quantities. The abstract maths of the standard approach (complex numbers and spinors and so forth) works, but you often find yourself just turning the handle and crossing your fingers, with no clue what it means. It's a bit like Searle's Chinese Room - blind manipulation of symbols without understanding. It would be nice to undertand what the symbols mean.

The big problem is that Geometric Algebra is not widely taught, and requires not only learning entirely new notation for everything from vector algebra up, but quite a different mindset. For those with a personal interest in having an intuitive geometric picture of what it all means, it's great. But it could be a huge distraction and source of confusion (not to mention a lot of extra effort) if you're trying to follow a standard university physics course using the standard notation. You have been warned!

Doran, Lasenby, et al's survey paper "Spacetime algebra and electron physics" is a pretty good in depth introduction to the subject as it is applied to quantum physics. (There are a lot of other papers around that go through the basics more gently, that I'd start with first. Just search on "Geometric Algebra" and "Spacetime Algebra")

In Geometric Algebra all the imaginary units are replaced with geometric objects/transformations that still have the property that they square to -1, but are really things like rotations. The Pauli matrices (which are Hermitian) are interpreted as an orthonormal basis of vectors in 3D space. The Dirac gamma matrices (one Hermitian, three anti-Hermitian) represent an orthonormal basis in 4D spacetime. When applied as operators to vectors of complex numbers, they are simply picking out $x$, $y$, $z$, and $t$ components of some geometrical structure like a line or a plane. Unit vectors can be interpreted as reflections and also oriented directions in space, bivectors (pairs of vectors) as rotations (pairs of reflections) and also oriented planes in space, trivectors as rotoreflections and volumes, and so on. The Hermitian conjugate operation corresponds to 'reversion' of the 3D spatial coordinates in a particular reference frame (reversing the order of reflections, having the effect of flipping various signs of directional components), so it can be used to separate spatial and temporal components of a geometrical structure, or express a 4D spacetime quantity in terms of 3D spatial vectors in a selected reference frame.

It has lots of other nice insights and simplifications, (and a whole new set of mysteries!) The four Maxwell equations of electromagnetism crunch down to a much simpler single equation. Relativistic manipulations are often easier. Much of geometry is simplified. Dot and cross products of vectors are unified into a single operation, and made associative and (usually) invertible. Div, grad, and curl are all combined together in one (usually) invertible operation. All the variants on Stokes theorem and the Cauchy integral formula are unified and generalised to higher dimensions. Spinors turn out to be just the even subalgebra of a Geometric Algebra (terms made up from even number of vectors multiplied together), and basically just a coordinate rotation. There are extensions to projective and conformal geometries that unify translations and rotations, planes and spheres, lines and circles, linear momentum and angular momentum, mass and moment of inertia, and so on. And the mysterious imaginary $i$ in Schrodinger's equation and Dirac's theory turns out to be just the spin plane.

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In QM we still want expectation values of our observables to be real numbers, i.e. we want $\langle\psi|\hat A|\psi\rangle$ to be real for observables. To see what this condition imposes on operators consider the expression $\langle\psi|\hat A|\phi\rangle$. Taking the complex conjugate gives \begin{align} \langle\psi|\hat A|\phi\rangle^*&=\langle\psi|\hat A|\phi\rangle^\dagger=\langle\phi|\hat A^\dagger|\psi\rangle \end{align} So \begin{align} \langle\psi|\hat A|\psi\rangle^*&=\langle\psi|\hat A|\psi\rangle\\ \implies \langle\psi|\hat A^\dagger|\psi\rangle&=\langle\psi|\hat A|\psi\rangle\quad\text{(for all $\psi$)}\\ \implies\hat A&=\hat A^\dagger \end{align}

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